Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light Andre´ Waser CH-8840 Einsiedeln, Switzerland; [email protected] The biquaternion notation - or more precisely the semi-biquaternion notation - is ap- plied to mechanics and special relativity. The Lorentz transformation as well as electro- magnetic field transformations result from a pure multiplication with the biquaternion relative velocity. By a straight forward development of the biquaternion notation, it turns out that the biquaternion velocity plays a crucial role regarding the principle of equivalence. The total time derivation of the four-velocity delivers inertial as well as gravitational acceleration simultaneously. It our model, the cause of gravitation is a spatial varying speed of light. Issued: January 01, 2011 1 Introduction The different brackets shall indicate that the two semi-biqua- ternions are not of identical structure. The multiplication of A Hamilton quaternion q [1] is an expansion of real numbersp by the Hamilton units i, j, k having the magnitude of −1 two biquaternions and forming a four-number as " ! # a ib A = a + b = 0 + 0 q = q0 + iq1 + jq2 + kq3 ; i~a ~b whereas q :::q are real numbers. The Hamilton units consti- " ! # 0 3 x iy tute the associative but not commutative multiplications X = x + y = 0 + 0 i~x ~y i2 = j2 = k2 = i jk = −1 ; is i j = k jk = i ki = j ; 2 1 i j = − ji jk = −k j ki = −ik : a x − b y + ~a · ~x − ~b · ~y 6 0 0 0 0 C AX = 46 ~ ~ AC + The real part of a quaternion is often called the scalar part i a0~x + x0~a + b0~y + y0b + ~a × ~y − b × ~x while the other three imaginary parts are called a three-vector. 0 ~ 3 B i a0y0 + b0 x0 − ~a · ~y − b · ~x 7 Both together constitute a four vector: @B 57 : (1) a0~y + x0~b − b0~x − y0~a − ~a × ~x + ~b × ~y q = q0; ~q ; The biquaternion multiplication is not commutative but asso- where ~q = iq1 + jq2 + kq3. Later Hamilton [2] presented the ciative. For the biquaternion scalar multiplication the Hamil- biquaternion as an extension of a quaternion by introducing ton units are not multiplied and thus complex numbers q ! q +ip , where i denotes the Gaussian n n n imaginary unit. A · X = a0 x0 − ~a · ~x − b0y0 − ~b · ~y : Q = q + ip + i (q + ip ) + j (q + ip ) + k (q + ip ) : 0 0 1 1 2 2 3 3 Thereof follows the magnitude of a biquaternion Generally a biquaternion can be grouped into two four-vec- p q tors jQj = Q · Q = a2 − ~a · ~a − b2 − ~b · ~b : (2) 0 0 Q = q0; i~q + ip0; ~p = q + p : Both terms q and p shall be called semi-biquaternions. These A special case is given when the second of the two semi- two semi-biquaternions do not have the same structure. The biquaternions is set to zero. With b = 0 and ~y = 0, the multi- first term is very suitable to cast translational physical four- plication of two biquaternions reduces to quantities whereas the second may represent a complemen- " ! # a x + ~a · ~x 0 tary rotational four-quantity, if existing. AX = 0 0 + : − × For a more convenient readability a biquaternion can be i a0~x + x0~a ~a ~x written as the sum of a ’left’ and ’right’ semi-biquaternion as The multiplication of a semi-biquaternion with an other one " ! # q ip results almost in another semi-biquaternion, except of an ad- Q = q + p = 0 + 0 : i~q ~p ditional cross product. Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light1 2 Biquaternion Velocity With equation (6) follows We have shown some time ago that such semi-biquaternions dX · dX V0 · V0 = (8) can be applied well to electrodynamics [3]. In this paper we dX0 · dX0 focus on mechanics, and especially on special relativity. We This directly leads to the biquaternion coordinate transforma- start with the position biquaternion representing an event in 0 space-time tion from S to S " ! # 0 0∗ ct 0 X = XV (9) X = + (3) i~x 0 as well as the backward transformation where the constant c is the speed of light. With X = X0V0 (10) " ! # cdt 0 dX = + Explicitly the forward and backward transformations are id~x 0 " ! # " ! # ct0 0 ct − ~v·~v 0 and with the proper time interval dτ X0 = + = γ c + (11) i~x0 0 i ~x − ~vt ~x×~v r c p · " ! # " 0 ~v·~v ! # 2 2 ~v ~v ct 0 ct + 0 d jXj c dt − d~x · d~x cdt − cdτ : c 0 = ( ) = 1 2 = X = + = γ 0 0 + ~x ×~v : (12) c i~x 0 i ~x + ~vt − c follows the biquaternion velocity Equation (11) and equation (12) reveal, that " ! # dX c 0 0 V = = γ + (4) ~x × ~v = 0 and ~x × ~v = 0 : dτ i~v 0 Above transformations are valid for the special case of colli- with 0 1 near vectors ~xk~v and ~x k~v only. The general case is found γ = q : by transforming this part of position vector X with a spatial − v2 1 c2 vector parallel to ~v and add the remaining part of X with a spatial vector orthogonal to ~v. With Independent of the magnitude of ~v, the magnitude of the bi- " ! # quaternion velocity is always c. Thus the unity biquaternion ct 0 velocity is Xk = ~x·~v ~v + " ! # i v v 0 0 1 0 V = γ ~v + (5) i c 0 the general forward transformation is or 0 0∗ 0∗ dX V X = X? + XkV = X − Xk 1 − V ; (13) = = V0 (6) djXj jVj or explicitly The inverse of a unity biquaternion velocity is identical to 2 ~v·~v 1 # its biconjugate, which in turn is again a unity biquaternion 6 γ ct − C 0 X0 = 6 c C + ; velocity 46i ~x − (1 − γ) ~x·~v ~v − γ~vt AC 0 1 ∗ v v = V0 (7) V0 whereas the general backward transformation is with " ! # 0 0 0 0 0 0 ∗ 1 0 X = X + X V = X − X 1 − V ; (14) V0 = γ + : ? k k −i~v 0 c or explicitly 3 Biquaternion Lorentz Transformation 2 ~v·~v 1 # 6 γ ct + c C 0 S shall be the inertial system where an observer is at rest. X = 6 0 C + : 46i ~x0 − (1 + γ) ~x ·~v ~v + γ~vt AC 0 Another inertial system S0 shall move with constant velocity v v ~v against S. The distance between two events in four-space is No new Lorentz transformation has been described above. invariant for both systems But it might be of some interest to see that the Lorentz trans- djXj = djX0j : formation can be described with a multiplication of the po- sition biquaternion with the velocity biquaternion of relative Also the square of these length elements is an invariant and movement. The component of the position biquaternion par- therefore allel to the velocity biquaternion is dX · dX = dX0 · dX0 = d2jXj = d2jX0j : X · V0 V0 : 2 Andre´ Waser, Biquaternion Relativity - Gravitation as an Effect of Spatial Varying Speed of Light Its Lorentz transformation is The electromagnetic fields follow from the differentiation of ! the potential fields. With the definition of the biquaternion ∗ ~x · ~v X · V0 V0V0 = γ ct − : differential operator c " ! # The result has no spatial position vector at all and has a pure @ 1 @ 0 r ≡ = c @t + : (17) timely character in all cases, i.e. independent of the direction @X r~ 0 of the unprimed position biquaternion or of its spatial vector. Thus, for every observer in S, the direction of the time vec- follow the biquaternion electric and magnetic fields 0 tor (scalar part) in any moving system S coincides with the 2 0 0 1 @' ~ ~ ! # direction V of the relative four-velocity. 6c 2 − r · A B 0 E = crA = 6 c @t B + ; To find the addition law of two biquaternion velocities 4 iE~ @ cB~ we denote the relative biquaternion velocity between two sys- tems S and S0 as V, and the velocity of a moving body mea- " ! 0 1 @' ~ ~ 3 0 Bi( 2 − r · A)7 0 0 B = −irA = + B c @t 7 ; sured in S shall be denoted as U . The total biquaternion iB~ @ − E~ 5 velocity W measured in S is found by the backward transfor- c mation of U0 as where E = −icB. With the Lorentz gauge " 0 # " # ~u ·~v 0 0 0 c + c 1 @' W = U V = γuγv 0 + γuγv ~u0×~v − r~ · A~ = 0 i ~u + ~v c c2 @t with the biquaternion electric and magnetic fields without a scalar 1 1 γu = q and γv = q : part are found 02 2 " ! # 1 − u 1 − v 0 0 c2 c2 E = + ; (18) iE~ cB~ Some calculation directly leads to the total velocity " ! # " ! # c 0 0 0 B = + ~ : (19) W = γw + w~×~v iB~ − E iw~ − c c with With the electric field biquaternion (18), the forward transfor- 1 mation is γw = q 1 − w2 " ! # c2 0 0 0 E = 0 + 0 and the law of velocity addition known from Special Relativ- iE~ B~ 2 1 1 0 3 ity ∗ 6 − ~v · E~ C B~v · B~ 7 ~u0 + ~v = EV0 = γ 6 c C + γ B 7 : (20) w~ = : 4i E~ + ~v × B~ A @ − E~ 5 ~u0·~v c 1 + c2 Again, the magnitude of W is equal to the speed of light c. Obviously this transformation is only applicable for 4 Biquaternion Field Transformations ~v · E~ = 0 and ~v · B~ = 0 : The biquaternion electromagnetic potential Thus only the field components orthogonal to the spatial ve- " ! # 1 ' 0 locity vector are transformed, but not the collinear field com- A ≡ c + (15) A~ 0 ponents.